Question: Determine how many solutions exist for the system of equations. ${-12x+3y = 21}$ ${y = 7+4x}$
Solution: Convert both equations to slope-intercept form: ${-12x+3y = 21}$ $-12x{+12x} + 3y = 21{+12x}$ $3y = 21+12x$ $y = 7+4x$ ${y = 4x+7}$ ${y = 7+4x}$ ${y = 4x+7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+7}$ ${y = 4x+7}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-12x+3y = 21}$ is also a solution of ${y = 7+4x}$, there are infinitely many solutions.